<p>Let A = {2, 3, 4, . . . , 30} and “∼” be defined on A × A by (a, b) ∼(c, d) iff ad = bc. The
number of ordered pairs which are related to (4, 3) is:</p>
Step-by-Step Solution
Key Concept: ad = bc \Leftrightarrow a
b = c
d. So (c, d) ∼(4, 3) iff c
d = 4
3, i.e. c = 4k, d = 3k for some k \in N.
<p><strong>Step 1</strong>: Find valid k: need 4k \in A and 3k \in A, i.e. 2 \leq4k \leq30 and 2 \leq3k \leq30. Both conditions give</p><br>k \in {1, 2, 3, 4, 5, 6, 7}.<p><strong>Step 2</strong>: List all 7 pairs in the equivalence class:</p><br>k<br>(4k, 3k)<br>In A \times A?<br>1<br>(4, 3)<br>✓<br>2<br>(8, 6)<br>✓<br>3<br>(12, 9)<br>✓<br>4<br>(16, 12)<br>✓<br>5<br>(20, 15)<br>✓<br>6<br>(24, 18)<br>✓<br>7<br>(28, 21)<br>✓<p><strong>Step 3</strong>: The question asks for all pairs related to (4, 3), which includes (4, 3) itself. Total = 7.</p>
Correct Answer: 4